Semiparametric Fisher Information in Models parametrized by a Normed Space

TL;DR

This paper generalizes Fisher information theory to models parametrized by normed spaces, establishing conditions for positive information and linking differentiability with information in a unified framework.

Contribution

It extends key theorems to normed spaces, providing a unified approach to differentiability and Fisher information beyond Hilbert spaces.

Findings

Positive Fisher information is equivalent to the gradient lying in the range of the adjoint score operator.

Finite variance of the transformation is necessary for positive information in mean-square-differentiable models.

Positive mass at the evaluation point is necessary for positive information in density estimation.

Abstract

This paper studies semiparametric Fisher information in models parametrized by general normed spaces. The main contribution is to establish that positive semiparametric Fisher information is equivalent to the gradient of the parameter of interest lying in the range of the adjoint score operator. This result generalizes a key theorem Van Der Vaart (1991) and provides a unified framework linking differentiability and information, beyond Hilbert spaces. The paper develops a normed-space mean-square-differentiable models for two canonical problems: estimation of the average of a known transformation and estimation of a density at a point. In these applications, it shows that positive information holds if and only if the transformation has finite variance and if and only if the density has positive mass at the evaluation point, respectively. These findings offer a novel information-theoretic perspective on known minimax results and clarify the conditions under which root-n estimation is possible.